Analytic Feynman Integral and a Change of Scale Formula for Wiener Integrals of an Unbounded Cylinder Function

Abstract

We investigate the behavior of the unbounded cylinder function $F\left(x\right)={\left({\displaystyle {\int }_0^T{\alpha }_1\left(t\right)\text{d}x\left(t\right)}\right)}^{2k}\cdot {\left({\displaystyle {\int }_0^T{\alpha }_2\left(t\right)\text{d}x\left(t\right)}\right)}^{2k}\cdot \cdots \cdot {\left({\displaystyle {\int }_0^T{\alpha }_n\left(t\right)\text{d}x\left(t\right)}\right)}^{2k},k=\mathrm{1,2},\dots$ whose analytic Wiener integral and analytic Feynman integral exist, we prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral, and we prove a change of scale formula for the Wiener integral about the unbounded function on the Wiener space $C_0\left[0,T\right]$ .